Algorithms and Orders for Finding Noncummutative Grobner Bases

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Abstract

The problem of choosing efficient
algorithms and good admissible
orders for computing Gröbner bases
in noncommutative algebras is
considered. Gröbner bases are an
important tool that make many
problems in polynomial algebra
computationally tractable. However,
the computation of Gröbner bases is
expensive, and in noncommutative
algebras is not guaranteed to
terminate. The algorithm, together
with the order used to determine the
leading term of each polynomial, are
known to affect the cost of the
computation, and are the focus of
this thesis.
A Gröbner basis is a set of
polynomials computed, using
Buchberger's algorithm, from another
set of polynomials. The
noncommutative form of
Buchberger's algorithm repeatedly
constructs a new polynomial from a
triple, which is a pair of polynomials
whose leading terms overlap and
form a nontrivial common multiple.
The algorithm leaves a number of
details underspecified, and can be
altered to improve its behavior. A
significant improvement is the
development of a dynamic dictionary
matching approach that efficiently
solves the pattern matching problems
of noncommutative Gröbner basis
computations. Three algorithmic
alternatives are considered: the
strategy for selecting triples
(selection), the strategy for removing
triples from consideration (triple
elimination), and the approach to
keeping the set interreduced (set
reduction).
Experiments show that the selection
strategy is generally more significant
than the other techniques, with the
best strategy being the one that
chooses the triple with the shortest
common multiple. The best triple
elimination strategy ignoring resource
constraints is the Gebauer-Müller
strategy. However, another strategy
is defined that can perform as well as
the Gebauer-Müller strategy in less
space.
The experiments also show that the
admissible order used to determine
the leading term of a polynomial is
more significant than the algorithm.
Experiments indicate that the choice
of order is dependent on the input
set of polynomials, but also suggest
that the length lexicographic order is
a good choice for many problems. A
more practical approach to chosing
an order may be to develop
heuristics that attempt to find an
order that minimizes the number of
overlaps considered during the
computation.